# Itô’s formula

## 0.1 Case of single space dimension

Let $X_{t}$ be an Itô process satisfying the stochastic differential equation

 $dX_{t}=\mu_{t}\,dt+\sigma_{t}\,dW_{t}\,,$

with $\mu_{t}$ and $\sigma_{t}$ being adapted processes, adapted to the same filtration as the Brownian motion $W_{t}$. Let $f$ be a function with continuous partial derivatives $\frac{\partial f}{\partial t}$, $\frac{\partial f}{\partial x}$ and $\frac{\partial^{2}f}{\partial x^{2}}$.

Then $Y_{t}=f(X_{t})$ is also an Itô process, and its stochastic differential equation is

 $\displaystyle dY_{t}$ $\displaystyle=\frac{\partial f}{\partial t}\,dt+\frac{\partial f}{\partial x}% \,dX_{t}+\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}(dX_{t})(dX_{t})$ $\displaystyle=\left(\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x% }\mu_{t}+\frac{1}{2}\sigma_{t}^{2}\right)\,dt+\frac{\partial f}{\partial x}% \sigma_{t}\,dW_{t}\,,$

where all partial derivatives are to be taken at $(t,X_{t})$.

## 0.2 Case of multiple space dimensions

There is also an analogue for multiple space dimensions.

Let $X_{t}$ be a $\mathbb{R}^{n}$-valued Itô process satisfying the stochastic differential equation

 $dX_{t}=\mu_{t}\,dt+\sigma_{t}\,dW_{t}\,,$

with $\mu_{t}$ and $\sigma_{t}$ being adapted processes, adapted to the same filtration as the $m$-dimensional Brownian motion $W_{t}$. $\mu_{t}$ is $\mathbb{R}^{n}$-valued and $\sigma_{t}$ is $L(\mathbb{R}^{m},\mathbb{R}^{n})$-valued.

Let $f\colon\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ be a function with continuous partial derivatives.

Then $Y_{t}=f(X_{t})$ is also an Itô process, and its stochastic differential equation is

 $\displaystyle dY_{t}$ $\displaystyle=\frac{\partial f}{\partial t}\,dt+(\operatorname{D}f)\,dX_{t}+% \tfrac{1}{2}dX_{t}^{*}(\operatorname{D}^{2}f)dX_{t}$ $\displaystyle=\frac{\partial f}{\partial t}\,dt+(\operatorname{D}f)\mu_{t}\,dt% +(\operatorname{D}f)\sigma_{t}\,dW_{t}+\tfrac{1}{2}dW_{t}^{*}\,\sigma_{t}^{*}(% \operatorname{D}^{2}f)\sigma_{t}\,dW_{t}$ $\displaystyle=\frac{\partial f}{\partial t}\,dt+(\operatorname{D}f)\mu_{t}\,dt% +(\operatorname{D}f)\sigma_{t}\,dW_{t}+\tfrac{1}{2}\operatorname{tr}\bigl{(}% \sigma_{t}^{*}\,(\operatorname{D}^{2}f)\,\sigma_{t}\bigr{)}\,dt$ $\displaystyle=\left(\frac{\partial f}{\partial t}+(\operatorname{D}f)\mu_{t}+% \tfrac{1}{2}\operatorname{tr}\bigl{(}(\sigma_{t}\sigma_{t}^{*})(\operatorname{% D}^{2}f)\bigr{)}\right)\,dt+(\operatorname{D}f)\sigma_{t}\,dW_{t}\,,$

where

• $\operatorname{tr}$ is the trace operation; ${}^{\ast}$ is the transpose

• $\operatorname{D}f$ is the derivative with respect to the space variables; its value is a linear transformation from $L(\mathbb{R}^{n},\mathbb{R})$

• $\operatorname{D}^{2}f$ is the second derivative with respect to space variables; represented as the Hessian matrix

• the third line follows because $dW_{t}^{i}\,dW_{t}^{j}=\delta_{ij}\,dt$.

The quadratic form $\operatorname{tr}(\sigma_{t}\sigma_{t}^{*}\,\operatorname{D}^{2}f)\,dt$ represents the quadratic variation of the process. When $\sigma_{t}$ is the identity transformation, this reduces to the Laplacian of $f$.

Itô’s formula in multiple dimensions can also be written with the standard vector calculus operators. It is in the similar notation typically used for the related parabolic partial differential equation describing an Itô diffusion:

 $dY_{t}=\left(\frac{\partial f}{\partial t}+\mu_{t}\cdot\nabla f+\tfrac{1}{2}% \bigl{(}\nabla\cdot(\sigma_{t}\sigma_{t}^{*})\nabla\bigr{)}f\right)dt+(\sigma_% {t}\,dW_{t})\cdot\nabla f\,.$

## References

• 1 Bernt Øksendal. , An Introduction with Applications. 5th ed., Springer 1998.
• 2 Hui-Hsiung Kuo. Introduction to Stochastic Integration. Springer 2006.
 Title Itô’s formula Canonical name ItosFormula Date of creation 2013-03-22 17:16:14 Last modified on 2013-03-22 17:16:14 Owner stevecheng (10074) Last modified by stevecheng (10074) Numerical id 13 Author stevecheng (10074) Entry type Axiom Classification msc 60H10 Classification msc 60H05 Synonym Itô’s formula Synonym Itô’s chain rule Synonym Ito’s formula Synonym Ito’s lemma Synonym Ito’s chain rule Related topic ItosLemma2 Related topic GeneralizedItoFormula